midterm09sols

# midterm09sols - L. Vandenberghe 10/27/09 EE103 Midterm...

This preview shows pages 1–3. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: L. Vandenberghe 10/27/09 EE103 Midterm Solutions Problem 1 (10 points). The cross-product a × x of two 3-vectors a = ( a 1 ,a 2 ,a 3 ) and x = ( x 1 ,x 2 ,x 3 ) is defined as the vector a × x = a 2 x 3 − a 3 x 2 a 3 x 1 − a 1 x 3 a 1 x 2 − a 2 x 1 . 1. Assume a is fixed and nonzero. Show that the function f ( x ) = a × x is a linear function of x , by giving a matrix A that satisfies f ( x ) = Ax for all x . 2. Is the matrix A you found in part 1 singular or nonsingular? 3. Verify that A T A = ( a T a ) I − aa T . 4. Use the observations in parts 1 and 3 to show that for nonzero x , bardbl a × x bardbl = bardbl a bardblbardbl x bardbl| sin θ | where θ is the angle between a and x . Solution. 1. a × x = Ax with A = − a 3 a 2 a 3 − a 1 − a 2 a 1 . 2. Singular, because Aa = 0 so A has a nonzero vector in its nullspace. 3. Working out the matrix product gives A T A = a 2 2 + a 2 3 − a 1 a 2 − a 1 a 3 − a 1 a 2 a 2 1 + a 2 3 − a 2 a 3 − a 1 a 3 − a 2 a 3 a 2 1 + a 2 3 = a 2 1 + a 2 2 + a 2 3 a 2 1 + a 2 2 + a 2 3 a 2 1 + a 2 2 + a 2 3 − a 2 1 a 1 a 2 a 1 a 3 a 1 a 2 a 2 2 a 2 a 3 a 1 a 3 a 2 a 3 a 2 3 = ( a T a ) I − aa T . 1 4. From the expression in part 3, bardbl a × x bardbl 2 = x T A T Ax = x T (( a T a ) I − aa T ) x = ( a T a )( x T x ) − ( x T a ) 2 = bardbl a bardbl 2 bardbl x bardbl 2 − ( bardbl a bardblbardbl x bardbl cos θ ) 2 = ( bardbl a bardblbardbl x bardbl sin θ ) 2 ....
View Full Document

## This note was uploaded on 02/03/2011 for the course EE 103 taught by Professor Vandenberghe,lieven during the Spring '08 term at UCLA.

### Page1 / 6

midterm09sols - L. Vandenberghe 10/27/09 EE103 Midterm...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online